\section{Construction}
\label{sec:construction}
 Let $x\in L_1$, and let $y_0\in L_1$ be the first point we choose. Since we want to minimize the forbidden area, we must place the next point, $y_1$, anywhere on the intersection of the circles that are the intersections of the sphere $S$ and the planes. We want to choose the third point in such a way that the forbidden area is as small as possible, so we will place the point somewhere in the circle that is the intersection of the sphere $S$ and the plane that is perpendicular to the segment $(x,y_1)$ and cuts it in half.

 In order to minimize the forbidden space, we need to place the new points as close as possible to the points that have already been placed, so we will keep adding points in the intersections of the circles that are the intersection of $S$ and the segments between $x$ and the points we have.

 Everytime we add a new point, a new circle emerges, that intersects the circles that were already there, so we can always keep placing points in the intersections of the cirles, we are finished when there is no more space to add any more points, now we shall see how many points are after we are done.